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XXII OM - III - Problem 5
Find the largest integer $ A $ such that for every permutation of the set of natural numbers not greater than 100, the sum of some 10 consecutive terms is at least $ A $. | The sum of all natural numbers not greater than $100$ is equal to $1 + 2 + \ldots + 100 = \frac{1 + 100}{2} \cdot 100 = 5050$. If $a_1, a_2, \ldots, a_{100}$ is some permutation of the set of natural numbers not greater than $100$ and the sum of any $10$ terms of this permutation is less than some number $B$, then in p... | 505 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
L OM - II - Task 4
Point $ P $ lies inside triangle $ ABC $ and satisfies the conditions: $ \measuredangle PAB = \measuredangle PCA $ and $ \measuredangle PAC = \measuredangle PBA $. Point $ O $ is the center of the circumcircle of triangle $ ABC $. Prove that if $ O \ne P $, then angle $ APO $ is a right angle. | Let $ K $, $ L $, $ M $ be the points of intersection of the lines $ AP $, $ BP $, $ CP $ with the circumcircle of triangle $ ABC $. By the equality $ \measuredangle BAK = \measuredangle ACM $, the lengths of the arcs $ BK $ and $ AM $ are equal. Similarly, the lengths of the arcs $ KC $ and $ LA $ are equal. The segme... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXXIII OM - I - Problem 9
In a chessboard created by dividing a square of side length $ n $ into unit squares using lines parallel to the sides of the square, we consider all squares whose sides are contained in the lines forming the chessboard. Let $ 1 \leq k \leq n $ and $ P(k,n) $ denote the number of these squares... | On the side of the chessboard, there are $ n - k + 1 $ segments of length $ k $ with endpoints at the division points. Hence, the total number of squares of side length $ k $ contained in the lines forming the chessboard is $ (n - k + 1)^2 $. Therefore, we have the following formulas:
The number $ k(n) $ satisfies the... | \frac{\sqrt[3]{4}}{2} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
XLVIII OM - III - Problem 5
Given is a convex pentagon $ABCDE$, where $DC = DE$ and $\measuredangle DCB = \measuredangle DEA = 90^\circ$. Let $F$ be the point on side $AB$ determined by the condition $AF: BF = AE: BC$. Prove that $\measuredangle FCE = \measuredangle ADE$ and $\measuredangle FEC = \measuredangle BDC$. | On the extension of segment $ BC $, we lay down segment $ CP = AE $. From the conditions of the problem, it follows that the right triangles $ PCD $ and $ AED $ are congruent. Therefore, $ AD = DP $ and
Since also $ ED = DC $, triangles $ ADP $ and $ EDC $ are similar. Denoting by $ Q $ the intersection point of lin... | proof | Geometry | proof | Yes | Yes | olympiads | false |
IX OM - III - Task 3
Prove that if $ n $ is a natural number greater than $ 1 $, then | \spos{1} Let's consider $ n $ unit vectors $ \overrightarrow{OA_1}, \overrightarrow{OA_2}, \ldots, \overrightarrow{OA_{n-1}}, \overrightarrow{OA_n} $ forming angles with the $ OX $ axis of $ \frac{2\pi}{n}, \frac{4\pi}{n}, \ldots, \frac{(2n-2) \pi}{n}, \frac{2n \pi}{n} $ (Fig. 29). Let the sum of these vectors be some... | proof | Number Theory | proof | Incomplete | Incomplete | olympiads | false |
IV OM - I - Problem 6
Prove that if a plane figure has two and only two axes of symmetry, then these axes are perpendicular. | The theorem will be proven when we show that if a plane figure has two axes of symmetry that are not perpendicular, then it has at least one more axis of symmetry.
Suppose that the plane figure $F$ has two axes of symmetry $k$ and $l$, which are not perpendicular. They can intersect (Fig. 26) or be parallel (Fig. 27). ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
III OM - II - Task 3
Are the statements true:
a) if four vertices of a rectangle lie on the four sides of a rhombus, then the sides of the rectangle are parallel to the diagonals of the rhombus;
b) if four vertices of a square lie on the four sides of a rhombus, which is not a square, then the sides of the square are ... | a) The first theorem is not true; to demonstrate this, it suffices to provide a counterexample, i.e., to show a figure contradicting the theorem.
From the center $ O $ of the rhombus $ ABCD $ (Fig. 28), draw a circle with a radius greater than the distance from the center of the rhombus to its side, but smaller than ha... | proof | Geometry | proof | Yes | Yes | olympiads | false |
LVIII OM - I - Problem 3
In a convex quadrilateral $ABCD$, which is not a parallelogram, the equality $AB=CD$ holds. Points $M$ and $N$ are the midpoints of diagonals $AC$ and $BD$, respectively. Prove that the orthogonal projections of segments $AB$ and $CD$ onto line $MN$ are segments of equal length, equal to the l... | Let point $E$ be the midpoint of side $AD$, and point $F$ be the midpoint of segment $MN$ (Fig. 1).
om58_1r_img_1.jpg
From the converse of Thales' theorem, it follows that $EM \parallel DC$ and $EM = \frac{1}{2}DC$. Similarly, we obtain the relationships $EN \parallel AB$ and $EN = \frac{1}{2}AB$. From the equality $DC... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXVIII - I - Problem 10
Prove that in every tetrahedron there exists a vertex at which all three dihedral angles are acute. | The sum of the measures of the plane angles of any face of a tetrahedron is equal to $ \pi $. Therefore, the sum of the measures of all the plane angles of the tetrahedron is $ 4\pi $. It follows that there exists a vertex $ A $ of the tetrahedron such that the sum of the measures of the plane angles at this vertex doe... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXXI - I - Problem 9
Prove that a natural number $ n > 1 $ divides the number $ a^{(n-1)! + 1} - a $ for every integer $ a $ if and only if $ n $ is a product of distinct prime numbers. | Integers that are not divisible by the square of any prime number are called square-free numbers. First, we will show that the condition for \( n \) to be a square-free number is necessary for \( n \) to divide \( a^{(n-1)!+1} - a \) for every integer \( a \). Suppose \( n \) is divisible by the square of a prime numbe... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
LIX OM - II - Task 5
Given is a triangle $ ABC $, in which $ AC = BC $. Point $ D $ lies on side $ AB $ of this triangle, with $ AD < BD $. Point
$ E $ is symmetric to point $ A $ with respect to the line $ CD $. Prove that | Let $ F $ be the point symmetric to $ A $ with respect to point $ D $ (Fig. 3). By the inequality $ AD < BD $, point $ F $ lies inside segment $ AB $.
Since $ AD = DF $ and points $ A $ and $ E $ are symmetric with respect to line $ CD $, applying the converse of Thales' theorem, we conclude that $ CD \parallel FE $. ... | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
XV OM - II - Task 3
Prove that if three prime numbers form an arithmetic progression with a difference not divisible by 6, then the smallest of these numbers is $3$. | Suppose that the prime numbers $ p_1 $, $ p_2 $, $ p_3 $ form an arithmetic progression with a difference $ r > 0 $ not divisible by $ 6 $, and the smallest of them is $ p_1 $. Then
Therefore, $ p_1 \geq 3 $, for if $ p_1 = 2 $, the number $ p_3 $ would be an even number greater than $ 2 $, and thus would not be a pri... | 3 | Number Theory | proof | Yes | Yes | olympiads | false |
XXVI - II - Problem 4
Prove that for non-negative numbers $ a_1, a_2, \ldots, a_n $ ($ n = 1, 2, \ldots $) to satisfy for any real numbers $ x_1, x_2, \ldots, x_n $ the inequality
it is necessary and sufficient that $ \sum_{i=1}^n a_i \leq 1 $. | If inequality (1) holds for any real numbers $ x_1, x_2, \ldots, x_n $, then in particular it holds for $ x_1 = x_2 = \ldots = x_n = 1 $. Substituting these values into (1) we obtain
Hence, given $ \displaystyle \sum_{i=1}^n a_i \geq 0 $, it follows that $ \displaystyle \sum_{i=1}^n a_i \leq 1 $.
For the proof of the ... | proof | Inequalities | proof | Incomplete | Yes | olympiads | false |
XIV OM - III - Task 4
Prove that for every natural number $ n $ the following inequality holds: | Let's denote for brevity $ 1 + \frac{1}{\sqrt{2}} + \ldots + \frac{1}{\sqrt{n}}= S_n $. We need to prove that $ S_n > \sqrt{n - 1} $ for every natural number $ n $. The idea of using complete induction comes to mind.
The theorem is true for $ n = 1 $, since $ S_1 = 1 > 0 $. Suppose that $ S_k > \sqrt{k-1} $ for some na... | proof | Inequalities | proof | Incomplete | Yes | olympiads | false |
XLIV OM - I - Problem 3
Given a hexagon $ABCDEF$ with a center of symmetry. Let $A$ be the intersection point of the lines $AB$ and $EF$, $B$ be the intersection point of the lines $BC$ and $AF$, and $C$ be the intersection point of the lines $AB$ and $CD$. Prove that $|AB| \cdot |BC| \cdot |CD| = |AA|. | A regular hexagon has a center of symmetry. Its opposite sides are therefore parallel and of equal length; in particular, the following relationships hold:
Since points $A$ and $B$ lie on the lines $EF$ and $BC$ respectively, and points $B$ and $C$ lie on the lines $FA$ and $CD$ respectively, we can rewrite the p... | proof | Geometry | proof | Incomplete | Incomplete | olympiads | false |
Given a positive integer $ n $ and positive real numbers $ a $, $ b $. Determine the maximum possible value of the sum
when $ x_{1},\ x_{2},\ \ldots,\ x_{n},\ y_{1},\ y_{2},\ \ldots,\ y_{n} $ are numbers from the interval $ \langle 0;1\rangle $, satisfying the conditions
Note: The original text had sever... | For any numbers $ x_{1},\ x_{2},\ \ldots,\ x_{n},\ y_{1},\ y_{2},\ \ldots,\ y_{n} $ satisfying the conditions of the problem, the following inequalities hold
Thus
We will first show that if $ a,\ b\geq n $ or $ [a]\neq[b] $ (the symbol $ [x] $ denotes the greatest integer not greater than $ x $), then the number ... | A+(-A)(b-A) | Inequalities | math-word-problem | Incomplete | Yes | olympiads | false |
XL OM - II - Task 5
Given is a sequence $ (c_n) $ of natural numbers defined recursively: $ c_1 = 2 $, $ c_{n+1} = \left[ \frac{3}{2}c_n\right] $. Prove that there are infinitely many even numbers and infinitely many odd numbers among the terms of this sequence. | It is enough to prove the following two statements:
(a) For any even number $ p $ being a term of the sequence $ (c_n) $, there exists a larger odd number that is a term of this sequence.
(b) For any odd number $ q $ being a term of the sequence $ (c_n) $, there exists a larger even number that is a term of this sequen... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XVIII OM - III - Problem 5
Prove that if a polygon with an odd number of sides inscribed in a circle has all angles equal, then the polygon is regular. | Assume that a polygon $ W $ with consecutive vertices $ A_1, A_2, \ldots, A_n $, where $ n $ is an odd number, has all angles equal and is inscribed in a circle $ O(r) $. When $ n = 3 $, the thesis of the theorem is obvious, so we will assume that $ n \geq 5 $.
Let $ A_{i-1}, A_i, A_{i+1} $ be three consecutive vertice... | proof | Geometry | proof | Yes | Yes | olympiads | false |
I OM - B - Task 19
The centers of four identical spheres lie on the circumference of a circle, and the center of mass of these spheres is at the center of the circle. Show that the centers of the spheres are the vertices of a rectangle. | Let $A, B, C, D$ denote the consecutive vertices of a quadrilateral formed by the centers of the spheres, and $M$ and $N$ - the midpoints of two opposite sides $AB$ and $CD$ of this quadrilateral.
Replace the spheres with centers $A$ and $B$ with a sphere twice as heavy with center $M$, and the spheres with centers $C$... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XLIX OM - III - Zadanie 2
Ciąg Fibonacciego $ (F_n) $ jest dany wzorami:
Wyznaczyć wszystkie pary $ (k, m) $ liczb całkowitych $ m > k \geq 0 $, dla których w ciągu $ (x_n) $ określonym wzorami
występuje liczba $ 1 $.
|
Załóżmy, że dla pewnej liczby całkowitej $ r \geq 0 $ zachodzi równość: $ x_r = 1 $, przy czym $ r $ jest najmniejszym numerem o tej własności. Jeśli $ r = 0 $, to $ F_k = F_m $, skąd $ k = 0 $, $ m = 1 $. Dalej zakładamy, że $ r \geq 1 $. W podanym wzorze rekurencyjnym podstawiamy $ n = r - 1 $ i wyznaczamy $ x_{r-1}... | (k,)=(2r,2r+1) | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
XI OM - II - Task 4
Prove that if $ n $ is a non-negative integer, then the number
is divisible by $ 7 $. | \spos{1} We apply mathematical induction. When $ n = 0 $, the theorem is true, since $ 2^2 + 3 = 7 $. Suppose the theorem is true when $ n = k \geq 0 $, then $ 2^{k+2} = 7m + r $, $ 3^{2k+1} = 7l - r $, where $ m $, $ l $ and $ r $ are non-negative integers, and therefore
The theorem is thus true when $ n = k + 1 $... | proof | Number Theory | proof | Incomplete | Incomplete | olympiads | false |
XLVIII OM - I - Zadanie 11
Dana jest liczba naturalna $ m\geq 1 $ oraz wielomian $ P(x) $ stopnia dodatniego o współczynnikach całkowitych mający co najmniej trzy różne pierwiastki całkowite. Dowieść, że wielomian $ P(x) + 5^m $ ma co najwyżej jeden pierwiastek całkowity.
|
Załóżmy, że liczby całkowite $ a $, $ b $, $ c $ są różnymi pierwiastkami wielomianu $ P(x) $. Przypuśćmy, wbrew dowodzonej tezie, że wielomian $ P(x) + 5^m $ ma dwa różne pierwiastki całkowite $ u $ i $ v $. Można przyjąć, że $ u>v $.
Dla każdej pary różnych liczb całkowitych $ x_1 $, $ x_2 $ liczba $ P(x_1)-P(x_2... | proof | Algebra | proof | Yes | Yes | olympiads | false |
XXXVII OM - III - Problem 4
Determine the non-negative integers $ n $ for which there exists a polynomial $ f $ of degree $ n $ with real coefficients such that $ f(x) > f for every real $ x $.
Note: The condition "$ f(x) > f $" seems to be incomplete or incorrect. If you meant "$ f(x) > 0 $" or another specific cond... | We will show that the desired property is possessed by all even numbers (and only them).
If $ f $ is a polynomial of degree $ n > 0 $ with the given property, then
is also a polynomial of degree $ n $, where $ g (x) \geq 0 $ for $ x \in \mathbb{R} $. Since every polynomial of odd degree (with real coefficients) a... | proof | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
LII OM - II - Problem 3
Let $ n \geq 3 $ be a natural number. Prove that any polynomial of the form
where at least one of the real coefficients $ a_0, a_1, \ldots, a_{n-3} $ is different from zero, has fewer than $ n $ real roots (each root is counted as many times as its multiplicity). | Suppose the polynomial given in the problem has $ n $ real roots $ x_1, x_2, \ldots, x_n $. From Vieta's formulas, it follows that
Thus, $ x_1^2 + x_2^2 + \ldots + x_n^2 = a_{n-1}^2 - 2a_{n-2} = 0 $, from which $ x_1 = x_2 = \ldots = x_n = 0 $. However, the above equality implies $ a_0 = a_1 = \ldots = a_{n-3} = 0 $, ... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
XX OM - III - Task 3
Prove that an octagon, all of whose sides are equal and all side lengths are rational numbers, has a center of symmetry. | According to a well-known theorem, the sum of the measures of the exterior angles of a polygon is $360^\circ$. Therefore, in an octagon $A_1A_2 \ldots A_8$ with equal angles, each exterior angle equals $45^\circ$.
Consider three consecutive sides of the octagon, for example, $A_1A_2$, $A_2A_3$, and $A_3A_4$. The extens... | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
XXI OM - II - Task 2
On the sides of a regular $ n $-gon, $ n + 1 $ points have been chosen, dividing the perimeter into equal parts. For which arrangement of the chosen points is the area of the convex polygon with these $ n + 1 $ vertices a) the largest, b) the smallest? | Let's denote by $a$ the $(n + 1)$-th part of the side length of the regular $n$-gon $A_1A_2 \ldots A_n$ (Fig. 10), i.e., the side length of this $n$-gon is $(n + 1)a$. Therefore, the perimeter of the $n$-gon is $n(n + 1)a$. Since $na < (n + 1)a < 2na$, at least one and at most two vertices of the $(n + 1)$-gon belong t... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false |
VII OM - I - Problem 5
Prove that if the real numbers $ p_1 $, $ p_2 $, $ q_1 $, $ q_2 $ satisfy the equation
then at least one of the equations
has real roots. | Denoting by $ \Delta_1 $ and $ \Delta_2 $ the discriminants of equations (2) we have $ \Delta_1 = p_1^2 -4q_1 $ $ \Delta_2 = p_2^2 - 4q_2 $, from which $ q_1 = \frac{1}{4} (p_1^2 - \Delta_1) $, $ q_2 = \frac{1}{4} (p_2^2 - \Delta_2) $. Substituting these values into equation (1), we obtain the equality
a hence
It tur... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
XXXIX OM - III - Problem 6
Calculate the maximum volume of a tetrahedron contained within a hemisphere of radius 1. | Let us denote the given hemisphere by $H$, the whole sphere by $K$, and the circle forming the flat part of the boundary of $H$ by $E$. Let $O$ be any tetrahedron contained in $H$. Its interior is the intersection of four open half-spaces determined by the planes of the faces. The center of the sphere $K$ does not belo... | \frac{\sqrt{3}}{4} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XXXVII OM - I - Problem 9
Prove that every natural number less than $ n! $ is the sum of at most $ n $ different natural numbers dividing $ n! $. | We prove by induction on $ n $. For $ n = 1 $, the theorem is satisfied "in a vacuum" (there are no natural numbers less than $ 1! $). Fix a natural number $ n $ and assume that the thesis of the theorem is true for it. We will prove the thesis for $ n + 1 $.
Take any natural number $ N < (n + 1)! $. We need to show th... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XXIII OM - I - Zadanie 11
Rozwiązać w liczbach naturalnych równanie
|
Zauważmy najpierw, że jeżeli $ n $ jest liczbą naturalną, a $ r $ - taką liczbą całkowitą, że
to liczba $ n^2 + r $ jest kwadratem pewnej liczby naturalnej tylko wtedy, gdy $ r = 0 $. Istotnie, z (1) wynika, że $ (n-1)^2 < n^2 + r < (n+1)^2 $, a liczba $ n^2 $ jest jedynym kwadratem liczby naturalnej zawar... | 3,11 | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
XLVII OM - I - Problem 7
Non-negative numbers $ a $, $ b $, $ c $, $ p $, $ q $, $ r $ satisfy the conditions:
Prove that $ 8abc \leq pa + qb + rc $ and determine when equality holds. | Without loss of generality, we can assume that $ a \geq b \geq c $. We will conduct a series of estimations based on the following inequalities (which are consequences of the assumption just made and the conditions of the problem):
Using the relationship (1), we get the estimation:
while using the relations... | proof | Inequalities | proof | Incomplete | Yes | olympiads | false |
XXIV OM - I - Problem 3
Let $ w(x) $ be a polynomial with integer coefficients. Prove that if a natural number $ d $ is a divisor of each of the numbers $ a_n = 3^n + w(n) $, where $ n = 0, 1, 2, \ldots $, then $ d $ is a power of 2 with an integer exponent. | Let the prime number $ p $ be a divisor of each of the numbers $ a_n $, where $ n = 0, 1, 2, \ldots $. It suffices to prove that $ p = 2 $.
In particular, we have $ p \mid a_0 $ and $ p \mid a_p $. Hence . Let $ w(x) = c_0 + c_1x + \ldots + c_rx^r $, where the numbers $ c_0, c_1, \ldots, c_r $ are integers. The number ... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XXXVIII OM - I - Problem 6
Which of the quadrilaterals having three sides of length 1 has the largest area? | We will use formula (1)
where
Formula (1) is valid for any quadrilateral, not necessarily convex; a proof can be found, for example, in S. Straszewicz's {\em Collection of Problems from Mathematical Olympiads} (PZWS 1956) in the commentary to the solution of problem 101 (p. 227).
Let us assume, in acc... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false |
VI OM - III - Task 2
Prove that among seven natural numbers forming an arithmetic progression with a difference of $30$, one and only one is divisible by $7$. | Seven numbers forming an arithmetic progression with a difference of $30$ can be written in the form: $a, a + 30, a + 2 \cdot 30, \ldots, a + 6 \cdot 30$. The difference between any two of these numbers has the form
\[
(k - m) \cdot 30
\]
where $k$ and $m$ are integers, with $0 \leq k \leq 6$, $0 \leq m \leq 6$, $k \... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
LI OM - III - Zadanie 6
Stopień wielomianu $ P(x) $ o współczynnikach rzeczywistych jest nieparzysty. Ponadto dla każdego $ x $
Udowodnić, że dla każdej liczby rzeczywistej $ x $ zachodzi równość
|
Dla dowolnej liczby rzeczywistej $ x $ prawdziwe są zależności
Stąd wynika, że dla każdego $ x $ zachodzi jedna z równości: $ P(-x) = P(x) $ lub P $ (-x) = -P (x) $.
Przypuśćmy, że dla nieskończenie wielu $ x $ zachodzi równość $ P(-x) = P(x) $. Wówczas równość ta musi być prawdziwa dla dowolnej liczby rzeczywis... | P(x)=x | Algebra | proof | Incomplete | Yes | olympiads | false |
VII OM - I - Problem 10
Prove that if $ \alpha > 0 $, $ \beta > 0 $, $ \gamma > 0 $ and $ \alpha + \beta + \gamma = \frac{1}{2} \pi $, then | From the assumption, it follows that $ \alpha $, $ \beta $, $ \gamma $, and $ \alpha + \beta = \frac{1}{2} \pi - \gamma $ are acute angles. Therefore,
hence,
and finally,
Note. The theorem can be generalized. For formula (1) to be true, it suffices to assume that each of the angles $ \alpha $, $ \beta... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXXIV OM - II - Problem 1
In a plane with a given coordinate system, there is a convex polygon whose all vertices have integer coordinates. Prove that twice the area of this polygon is an integer. | We choose any vertex of the polygon and draw all diagonals from it. In this way, we divide the given polygon into triangles, whose vertices have integer coordinates. It is therefore sufficient to prove the considered theorem for a triangle.
om34_2r_img_5.jpg
We draw lines through the vertices of the triangle parallel t... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXV - I - Problem 6
Prove that for any real numbers $ a, b, c $, a certain value of the function $ f(x) = |a \sin x + b \sin 2x + c \sin 4x| $ is not less than $ \frac{1}{2}(|a| + |b| + |c|) $. | Since changing the signs of the numbers $a$, $b$, $c$ does not change the conditions of the problem, we can assume without loss of generality that $a \geq 0$. Then, depending on the signs of the numbers $b$ and $c$, the following number $x$ satisfies the conditions of the problem: if $b \geq 0$ and $c \geq 0$, then $x ... | proof | Inequalities | proof | Yes | Incomplete | olympiads | false |
XXXV OM - I - Problem 4
Given are two intersecting circles $ o(O_1, r_1) $ and $ o(O_2, r_2) $ and a line tangent to them at points $ A $ and $ B $, respectively. Point $ C $ is the common point of these circles closer to the line $ AB $, $ R $ is the radius of the circumcircle of triangle $ ABC $, and $ d = O_1O_2 $.... | From the isosceles triangle $ AC0_1 $ we determine
Since $ \measuredangle O_1AB = 90^\circ $, then $ \cos O_1AC = \sin (90^\circ - O_1AC) = \sin BAC $. Applying the Law of Sines to triangle $ ABC $, we get
thus
Similarly,
From the last two equalities, it follows that
Calculating the sum of... | ^2=r_1^2+r_2^2+R^2 | Geometry | proof | Yes | Yes | olympiads | false |
IX OM - I - Task 2
Calculate the sum
| The $ k $-th term of the given sum equals
Substituting $ k = 1, 2, 3, \ldots, n $ in this equality, we get
[ 1 \cdot 2 \cdot 3 = 1^3 + 3 \cdot 1^2 + 2 \cdot 1, \]
[ 2 \cdot 3 \cdot 4 = 2^3 + 3 \cdot 2^2 + 2 \cdot 2, \]
[ 3 \cdot 4 \cdot 5 = 3^3 + 3 \cdot 3^2 + 2 \cdot 3, \]
[ \ldots \]
[ n (n + 1)... | notfound | Other | math-word-problem | Incomplete | Yes | olympiads | false |
XLVIII OM - III - Problem 4
The sequence $ a_1, a_2, a_3, \ldots $ is defined by the formulas
For every integer $ k \geq 0 $, determine the number of indices $ n $ satisfying the conditions
Note: $ [n/2] $ is the greatest integer not exceeding $ n/2 $.
| For each integer $ n \geq 1 $, we assign a finite sequence of two types of symbols: dots and commas. We write the number $ n $ in binary. Between each pair of consecutive digits, we insert a dot if the digits are the same, and a comma if they are different. If $ n $ satisfies the inequality $ 2^k \leq n < 2^{k+1} $, me... | {k\choose{k/2}} | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
XI OM - III - Task 5
From the digits $ 1 $, $ 2 $, $ 3 $, $ 4 $, $ 5 $, $ 6 $, $ 7 $, $ 8 $, $ 9 $, all possible four-digit numbers with different digits were formed. Find the sum of these numbers. | We will calculate how many four-digit numbers with different digits can be formed from the digits $1$ to $9$. The first digit $c_1$ can be chosen in $9$ ways; when $c_1$ is already set, the second digit $c_2$ can be chosen in $8$ ways; when $c_1$ and $c_2$ are set, there are $7$ possibilities for $c_3$; and when $c_1$,... | notfound | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
XXII OM - III - Problem 2
On a billiard table in the shape of a triangle, whose angles are proportional, a ball was struck from a certain internal point. The ball bounces off the walls according to the law "angle of incidence equals angle of reflection." Prove that the number of directions in which the ball can move i... | We will first prove the following
Lemma. After a ball is reflected twice in succession from the sides $\overline{CA}$ and $\overline{AB}$ of triangle $ABC$ (Fig. 14), the direction of the ball's movement changes by an angle of $2 \measuredangle BAC$.
Proof. Consider the image $AB'$ of triangle $ABC$ in the symmetry wit... | 2n | Geometry | proof | Yes | Yes | olympiads | false |
LII OM - III - Problem 2
Prove that the sum of the distances from any point lying inside a regular tetrahedron with edge length $1$ to its vertices is not greater than $3$. | Let $ A_1 A_2 A_3 A_4 $ be a given tetrahedron, and $ P $ any point in its interior. Suppose the line $ A_i P $ intersects the opposite face of the tetrahedron $ A_1 A_2 A_3 A_4 $ at point $ B_i $. Let $ \mathcal{C}_i $ be the tetrahedron whose one face is the face of the tetrahedron $ A_1 A_2 A_3 A_4 $ containing poin... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XLV OM - II - Task 4
Each vertex of a cube is assigned the number $ 1 $ or $ -1 $, and each face is assigned the product of the numbers assigned to the vertices of that face. Determine the set of values that the sum of the $ 14 $ numbers assigned to the faces and vertices can take. | Let's consider an arbitrary assignment of the numbers $1$ and $-1$ to the vertices of a cube; denote the sum of these numbers by $S_1$. Assign numbers to the faces according to a given rule; denote the sum of these numbers by $S_2$. The sum of all $14$ considered numbers is $S_1 + S_2$.
If we change the number at one v... | {14,10,6,2,-2,-6,-10,-14}\setminus{10,-14}={14,6,2,-2,-6,-10} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
XLVI OM - I - Zadanie 1
Wyznaczyć wszystkie pary $ (x,y) $ liczb naturalnych, dla których liczby $ \frac{x+1}{y} $ oraz $ \frac{y+1}{x} $ są naturalne.
|
Niech $ (x,y) $ będzie jedną z szukanych par. Liczby naturalne $ y \geq 1 $ oraz $ x \geq 1 $ są odpowiednio dzielnikami liczb $ x + 1 $ oraz $ y + 1 $. Zatem iloczyn $ xy $ jest dzielnikiem iloczynu $ (x + 1)(y+ 1) $, równego sumie $ xy + x + y + 1 $. Jest więc też dzielnikiem sumy $ x + y + 1 $, a zatem zachodzi nie... | (3,2),(2,3),(1,1),(2,1),(1,2) | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
XIV OM - III - Task 2
In space, there are four distinct points $ A $, $ B $, $ C $, $ D $. Prove that the three segments connecting the midpoints of segments $ AB $ and $ CD $, $ AC $ and $ BD $, $ AD $ and $ BC $ have a common midpoint. | It is enough to prove that any two of the three segments mentioned have a common midpoint. We will show that the midpoint $H$ of segment $MN$ coincides with the midpoint $K$ of segment $PQ$, where $M$, $N$, $P$, $Q$ denote the midpoints of segments $AB$, $CD$, $AC$, $BD$ respectively.
Consider the points $A$, $B$, $C$,... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XXX OM - III - Task 3
The experiment consists of performing $ n $ independent trials. The probability of a positive outcome in the $ i $-th trial is $ p_i $. Let $ r_k $ be the probability that exactly $ k $ trials yield a positive result. Prove that | Let $ \{1,2, \ldots,n\} = A_k \cup B_k $, where $ A_k = \{i_1, i_2, \ldots, i_k \} $ and $ B_k = \{i_{k+1}, i_{k+2}, \ldots, i_n \} $, be a partition of the set $ \{1, 2, \ldots, n \} $ into a $ k $-element set and a $ (n-k) $-element set.
The probability of the event that the trials numbered $ i_1, i_2, \ldots,i_k $ y... | proof | Combinatorics | proof | Yes | Yes | olympiads | false |
XVIII OM - I - Zadanie 10
Trójka liczb całkowitych tworzy postęp geometryczny o ilorazie całkowitym. Gdy najmniejszą z nich zwiększymy o 9, powstanie postęp arytmetyczny. Jakie to liczby?
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Poszukiwana trójka liczb ma postać $ (a, aq, aq^2) $, gdzie $ a $ i $ q $ są liczbami całkowitymi różnymi od zera. Rozróżnimy dwa przypadki.
a) Najmniejszą z liczb tej trójki jest $ a $ lub $ aq^2 $. Nową trójką liczb jest wtedy $ (a+9, aq, aq^2) $ lub $ (a, aq, aq^2+9) $; warunek, aby to był postęp arytmetyczny, wyra... | (-1,-4,-16),(-1,2,-4),(-9,-18,-36),(2,-4,8) | Algebra | math-word-problem | Yes | Yes | olympiads | false |
LV OM - I - Task 11
Point $ O $ is the center of the circle circumscribed around the isosceles trapezoid $ ABCD $ with bases $ AB $ and $ CD $. Points $ K $, $ L $, $ M $, $ N $ lie on sides $ AB $, $ BC $, $ CD $, $ DA $, respectively, and quadrilateral KLMN is a rhombus. Prove that point $ O $ lies on the line $ KM ... | Let $ P $, $ Q $, $ R $ be the midpoints of segments $ AD $, $ BC $, $ KM $ (Fig. 5). Then points $ P $, $ Q $, and $ R $ lie on a single straight line - parallel to the bases of trapezoid $ ABCD $. Draw a line through point $ L $ parallel to the bases of trapezoid $ ABCD $, intersecting segment $ AD $ at point $ S $.
... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XVII OM - II - Problem 4
Prove that if natural numbers $ a $ and $ b $ satisfy the equation $ a^2 + a = 3b^2 $, then the number $ a + 1 $ is a square of an integer. | Let the equality $ b = p_1^{\alpha_1} p_2^{\alpha_2} \ldots p_s^{\alpha_s} $ represent the prime factorization of the number $ b $. According to the assumption, the following equality holds:
Numbers $ a $ and $ a+1 $ are coprime, since any common divisor of these numbers is a divisor of the number $ (a+1)-a=1 $. Ther... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
LV OM - II - Task 4
Determine all positive integers $ n $ that have exactly $ \sqrt{n} $ positive divisors. | The number $ n = 1 $ satisfies the conditions of the problem. Let's assume, for the rest of the solution, that $ n > 1 $.
From the conditions of the problem, it follows that $ n = k^2 $ for some positive integer $ k $. The number of positive divisors of $ n $ that are less than $ k $ is equal to the number of its divis... | n=1n=9 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
XLI OM - I - Problem 5
Prove that the edges of a cube cannot be numbered with the numbers from 1 to 12 in such a way that the sum of the numbers of the edges emanating from each vertex is the same.
Can this condition be met by numbering the edges with twelve different numbers from the set $ \{1,2,\ldots,13\} $? | Suppose there exists a way of numbering as mentioned in the first sentence. Each edge is adjacent to two vertices at its two ends. Therefore, twice the sum of the numbers of all edges should be equal to $8s$, where $s$ represents the sum of the numbers of edges emanating from one vertex (a common value for all edges of... | proof | Combinatorics | proof | Yes | Yes | olympiads | false |
III OM - II - Task 1
Find the necessary and sufficient conditions that the real numbers $ a $, $ b $, $ c $ should satisfy so that the equation
has three real roots forming an arithmetic progression. | The roots of equation (1) form an arithmetic progression if and only if they can be represented in the form $ p - r $, $ p $, $ p + r $, where $ r \geq 0 $.
The case $ r = 0 $ means that equation (1) has three equal roots (a triple root); this occurs when the left side of equation (1) is the cube of a binomial of the f... | ^2-3b=02b^3-9abc+27c^2+27a^2c-7a^3=0 | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
Circles $ k_1 $, $ k_2 $, $ k_3 $, and $ k_4 $ are externally tangent: $ k_1 $ to $ k_2 $ at point $ A $, $ k_2 $ to $ k_3 $ at point $ B $, $ k_3 $ to $ k_4 $ at point $ C $, and $ k_4 $ to $ k_1 $ at point $ D $. Lines $ AB $ and $ CD $ intersect at point $ S $. A line $ p $ through point $ S $ is tangent to $ k_4 $ ... | Let $O_i$ be the center of the circle $k_i$, and $r_i$ be the length of its radius ($i = 1,2,3,4$). Thus,
om43_2r_img_8.jpg
We will show that
\begin{center}
(2) \qquad the closed broken line $O_1O_2O_3O_4O_1$ has no self-intersections.
\end{center}
Suppose, contrary to statement (2), that for example the segments... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXIX OM - III - Task 6
Prove that if in any tetrahedron $ h_1, h_2, h_3, h_4 $ are the lengths of the four heights, and $ d_1, d_2, d_3 $ are the distances between three pairs of opposite edges, then | Consider a tetrahedron $ABCD$. Let's consider the prism $ABCDEF$ with base $ABC$ and lateral edge $\overline{AD}$. The volume $V_S$ of this prism is three times the volume $V$ of the given tetrahedron. Moreover, $V = \frac{1}{3}h_1 \cdot S_{ABC}$, where $h_1$ is the length of the height of the tetrahedron drawn from ve... | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
XXVII OM - I - Zadanie 4
Samolot leci bez zatrzymywania się po najkrótszej drodze z Oslo do miasta $ X $ leżącego na równiku w Ameryce Południowej. Z Oslo startuje dokładnie w kierunku zachodnim. Wiedząc, że współrzędne geograficzne Oslo są : $ 59^{\circ}55 szerokości północnej i $ 10^{\circ}43 długości wschodniej, ob... |
Najkrótsza droga na powierzchni kuli łącząca dwa punkty tej kuli jest łukiem koła wielkiego, tzn. łukiem okręgu, którego środkiem jest środek kuli. Zatem samolot leci po łuku koła wielkiego przechodzącego przez Oslo. Przy tym Oslo jest najbardziej na północ wysuniętym punktem tego koła wielkiego, ponieważ samolot star... | Quito,10000 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XII OM - I - Zadanie 6
Dowieść, że jeżeli trzy liczby rzeczywiste $ a $, $ b $, $ c $, które nie są wszystkie równe zerw, spełniają warunki
to
|
Zauważmy najpierw, że jeżeli liczby $ a $, $ b $, $ c $ spełniają warunki (1) i (2) i któraś z nich równa się zeru, to na mocy (2) któraś z pozostałych też jest równa zeru, a wobec tego na mocy warunku (1) trzecia z tych liczb też równa jest zeru. Zatem warunek, że nie wszystkie $ 3 $ liczby $ a $, $ b $, $ c $ są rów... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
LII OM - III - Task 4
Given such integers $ a $ and $ b $ that for every non-negative integer $ n $ the number $ 2^na + b $ is a square of an integer. Prove that $ a = 0 $.
| If $ b = 0 $, then $ a = 0 $, because for $ a \ne 0 $, the numbers $ a $ and $ 2a $ cannot both be squares of integers.
If the number $ a $ were negative, then for some large natural number $ n $, the number $ 2^n a + b $ would also be negative, and thus could not be a square of an integer.
The only case left to consid... | 0 | Number Theory | proof | Yes | Incomplete | olympiads | false |
IX OM - I - Problem 4
Given a segment $ AH $ and a point $ M $ on it; construct a triangle $ ABC $ in such a way that the segment $ AH $ is one of the altitudes of the triangle $ ABC $, and the point $ M $ is the midpoint of one of the other altitudes. | Let $AH$ and $CL$ be the altitudes of triangle $ABC$, with $CM = ML$ (Fig. 4). Measure on $MA$ a segment $MN = HM$. Triangle $LMN$ is congruent to triangle $CMH$ ($ML = CM$, $MN = HM$, $\measuredangle LMN = \measuredangle CMH$), so $\measuredangle MNL = \measuredangle MHC = 90^\circ$. In the right triangle $ALM$, we kn... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XXXIV OM - I - Problem 11
Determine the number of subsets of the set $ \{1, 2, \ldots, n\} $ that do not contain two consecutive numbers. | For $ n= 1 $ there are two such subsets $ \emptyset $, $ \{1\} $, for $ n = 2 $ there are three: $ \emptyset $, $ \{1\} $, $ \{2\} $. Suppose that $ n > 2 $. Each subset of the set $ \{1, 2,\ldots, n+1\} $ not containing two consecutive numbers, to which the number $ n+1 $ does not belong, is a subset of the desired fo... | notfound | Combinatorics | math-word-problem | Yes | Incomplete | olympiads | false |
XXXV OM - I - Problem 2
Determine all values of the parameter $ a $ for which the system of equations
has solutions in real numbers. | Real numbers $ x $, $ y $, $ z $ are a solution to the given system of equations if and only if these numbers are roots of the polynomial
Indeed, numbers $ x $, $ y $, $ z $ are roots of the polynomial $ f $ if and only if
Therefore, the question posed in the problem reduces to finding for which values of $... | \in[0;\\frac{4}{27}] | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
VI OM - III - Task 3
A regular triangle $ABC$ is inscribed in a circle; prove that if $M$ is any point on the circle, then one of the distances $MA$, $MB$, $MC$ is equal to the sum of the other two. | When point $ M $ is at one of the vertices of triangle $ ABC $, the theorem is obvious, since in this case one of the distances $ MA $, $ MB $, $ MC $ equals zero. It remains to consider the case where $ M $ lies inside one of the arcs into which points $ A $, $ B $, $ C $ divide the circle, for example, inside the arc... | proof | Geometry | proof | Yes | Yes | olympiads | false |
VII OM - I - Problem 12
Prove that: a) the sum of the distances from any point $ M $ inside an equilateral triangle to its three sides is constant, i.e., it does not depend on the position of point $ M $ inside the triangle; b) the sum of the distances from any point inside a regular tetrahedron to its four faces is c... | \spos{1}
1°. Let us choose a point $ M $ inside an equilateral triangle $ ABC $ (Fig. 12). Let $ a $ denote the length of the side of this triangle, $ h $ - its height, $ x $, $ y $, $ z $ the distances from point $ M $ to the sides $ BC $, $ CA $, $ AB $, respectively. Since
then
from which
If the po... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXXI - I - Problem 5
A well-known mathematician, Professor X, is playing a game with his assistant, Master Y, as follows. X gives Y a finite sequence of vectors $ \overrightarrow{a_1}, \overrightarrow{a_2}, \ldots, \overrightarrow{a_n} $; all these vectors lie in the same plane and each has a length of 1. Knowing thes... | om31_1r_img_9.jpg
om31_1r_img_10.jpg
We will describe Igr's strategy inductively. $ \varepsilon_1 $ can be chosen arbitrarily, for example, $ \varepsilon_1 = 1 $. Then, $ \varepsilon_2 $ should be chosen so that the vector $ \varepsilon_1 \overrightarrow{a_1} + \varepsilon_2 \overrightarrow{a_2} $ has a length less tha... | proof | Combinatorics | proof | Yes | Yes | olympiads | false |
XXVII OM - I - Problem 6
Find the minimum value of the function defined on the set of real numbers by the following formula
where $ x_1, x_2, \ldots, x_n $ are given real numbers. | As is known, the following inequalities hold:
Therefore, for any real number $ t $ we have
Thus,
where $ t $ is any real number.
By the formula for the sum of the terms of a geometric series, we have
It follows that
for $ t \ne x_1 $. Therefore,
for $ t \ne x_1 $.
We have thus proved ... | x_1 | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
XII OM - I - Problem 8
Two equilateral triangles are inscribed in a circle. The vertices $ A $, $ B $, $ C $ of one triangle follow each other on the circle in the same direction as the vertices $ A_1 $, $ B_1 $, $ C_1 $ of the other triangle. The lines $ AB $ and $ A_1B_1 $ intersect at point $ P $, the lines $ BC $ ... | Let's rotate the entire figure (Fig. 8) around the center $ O $ of the given circle by $ 120^\circ $. Points $ A $, $ B $, $ C $ will move to the positions of points $ B $, $ C $, $ A $, respectively, and points $ A_1 $, $ B_1 $, $ C_1 $ will move to the positions of points $ B_1 $, $ C_1 $, $ A_1 $, respectively. The ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
LV OM - II - Task 6
At a party, $ n $ people met $ (n \geq 5) $. It is known that among any three people, some two know each other. Prove that among the party attendees, one can select at least $ n/2 $ people and seat them at a round table so that each person sits between two of their acquaintances. | Let $ X $ be the set of people present at the party. Let $ k $ be the largest natural number with the property: There exist $ k $ people $ A_1, A_2, \ldots, A_k $ such that for $ i = 1, 2, \ldots, k-1 $, people $ A_i $ and $ A_{i+1} $ know each other. Denote:
In the further solution, we will consider three cases:... | proof | Combinatorics | proof | Yes | Yes | olympiads | false |
The sequences $ (x_n) $ and $ (y_n) $ are defined as follows:
Prove that for every integer $ n\geq 0 $, the equality $ y_n = x_{2^n-1} $ holds. | We will deal with sequences $ (a_n) $, $ (b_n) $ with terms
The common value of $ a_0 $ and $ b_0 $ we denote by $ \lambda $:
From the recursive definitions of sequences $ (x_n) $ and $ (y_n) $ given in the problem, we derive analogous formulas for sequences $ (a_n) $ and $ (b_n) $:
Hence, by obvious induct... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
XVIII OM - I - Problem 12
Prove that if points $ A, B, C, D, E $ in space are positioned such that
then points $ A, B, C, D, E $ lie on the same plane. | Assume that points $A$, $B$, $C$, $D$, $E$ satisfy conditions (1) and (2). First, we will show that no two of these points can coincide. It suffices to prove that point $A$ cannot coincide with $B$ or with $C$, as any other case of coinciding points can be reduced to one of these two by changing the labels.
a) If point... | proof | Geometry | proof | Incomplete | Incomplete | olympiads | false |
XVII OM - II - Problem 6
Prove that the sum of the squares of the orthogonal projections of the sides of a triangle onto a line $ p $ in the plane of the triangle does not depend on the position of the line $ p $ if and only if the triangle is equilateral. | Let $ S $ denote the sum of the squares of the orthogonal projections of the sides of an equilateral triangle $ ABC $ with side length $ a $ onto a line $ p $ lying in the plane of the triangle. Since the projections of a segment onto parallel lines are equal, for the calculation of $ S $ we can assume that the line $ ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
LX OM - III - Task 4
Let $ x_1,x_2,\cdots ,x_n $ be non-negative numbers whose sum is 1. Prove that there exist
numbers $ a1,a2,\cdots ,an \in \{0, 1, 2, 3, 4\} $ such that $ (a_1,a_2,\cdots ,a_n) \neq (2, 2,\cdots , 2) $ and | Consider all possible sequences $ t =(t_1,t_2,\cdots ,t_n) $ such that $ t_1,t_2,\cdots ,t_n \in \{0, 1, 2\} $.
There are $ 3^n $ of them. For each of them, let $ S_t $ denote the sum $ t_1x_1 +t_2x_2 +\cdots +t_nx_n $.
Since the numbers $ x_1,x_2,\cdots ,x_n $ are non-negative and their sum is 1, the following inequal... | proof | Combinatorics | proof | Yes | Yes | olympiads | false |
XLV OM - II - Task 6
Given a prime number p. Prove the equivalence of the following statements:
(a) There exists an integer $ n $ such that the number $ n^2 - n + 3 $ is divisible by $ p $.
(b) There exists an integer $ m $ such that the number $ m^2 - m + 25 $ is divisible by $ p $. | The desired equivalence follows easily from the observation that
Implication (a) $ \Longrightarrow $ (b) is immediate: if (for some $ n $) the number $ n^2 -n + 3 $ is divisible by $ p $, it suffices to take $ m = 3n-1 $; according to (1), the number $ m^2 -m + 25 $ is also divisible by $ p $.
To prove the implication... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XLVI OM - I - Problem 3
A quadrilateral with sides $a$, $b$, $c$, $d$ is inscribed in a circle with radius $R$. Prove that if $a^2 + b^2 + c^2 + d^2 = 8R^2$, then one of the angles in the quadrilateral is a right angle or the diagonals are perpendicular. | Suppose that none of the interior angles of the considered quadrilateral $ABCD$ is a right angle; we need to show that in this case $AC \bot BD$. The quadrilateral is inscribed in a circle, so $|\measuredangle DAB| + |\measuredangle BCD| = 180^\circ$. Without loss of generality, assume that angle $DAB$ is acute, and an... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXX OM - III - Task 1
Given is a set $ \{r_1, r_2, \ldots, r_k\} $ of natural numbers, which when divided by a natural number $ m $ give different remainders. Prove that if $ k > \frac{m}{2} $, then for every integer $ n $ there exist such $ i $ and $ j $ ($ i $ and $ j $ not necessarily different), that the number $ ... | The set $ A $ of remainders when dividing $ n-r_1, n-r_2, \ldots, n-r_k $ by $ m $ contains more than $ \displaystyle \frac{m}{2} $ elements. Similarly, the set $ B $ of remainders when dividing $ r_1, r_2, \ldots, r_k $ by $ m $ also contains more than $ \displaystyle \frac{m}{2} $ elements. Sets $ A $ and $ B $ are c... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
LVI OM - I - Problem 6
Determine whether there exists an infinite sequence of natural numbers $ a_1,a_2, a_3,\ldots $ satisfying the equation
Please note that the equation is missing in the original text, so it is also missing in the translation. If you have the equation, please provide it, and I can includ... | We will show that the sequence $ (a_n) $ satisfying the conditions of the problem does not exist.
Suppose that the sequence $ (a_n) $ satisfies the conditions of the problem. We will reason in two ways.
From the given equality, it follows that the sequence $ (a_n) $ is increasing, and furthermore,
The number $ ... | proof | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
XL OM - III - Task 1
An even number of people sat down at a round table for the meeting. After the lunch break, the participants took their seats at the table in any order. Prove that there are two people who are separated by the same number of people as before the break. | Let's denote the number of people by $ n $. We number the seats at the table consecutively with integers from $ 0 $ to $ n - 1 $; seat number $ 0 $ and the direction of circulation are chosen arbitrarily. The person occupying seat $ k $ before the break takes seat $ m_k $ after the break; the sequence $ (m_0, \ldots, m... | proof | Combinatorics | proof | Yes | Yes | olympiads | false |
I OM - I - Task 1
Prove that if $ m > 0 $, then $ m + \frac{4}{m^2} \geq 3 $. | When we are looking for a proof of a theorem, we often proceed similarly to the analysis of a construction task, conducting so-called reductive reasoning.
We assume that the given theorem is true, draw conclusions from it, aiming to obtain some known true theorem. This true theorem is chosen as the starting point of th... | proof | Inequalities | proof | Yes | Incomplete | olympiads | false |
LIX OM - II - Task 3
Determine all functions $ f $ defined on the set of all real numbers and taking
real values, such that for any real numbers $ x $ and $ y $ the equality holds
untranslated part remains the same as it was not specified in the original text. | Let $ c = f(0) $. Substituting $ x = y = 0 $ in equation (1) we obtain the relation $ f(c)= f(f(0)) = 2f(0) = 2c $. Then
by setting $ x = 0 $ and $ y = c $ we observe that
It follows that $ c = 0 $, i.e., $ f(0) = 0 $.
Now, taking $ x = 0 $ in condition (1), we see that
Next, let us substitute $ y = f(x)... | f(x)=-x | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
XV OM - II - Problem 5
Given is a trihedral angle with edges $SA$, $SB$, $SC$, where all dihedral angles are acute, and the dihedral angle along the edge $SA$ is right. Prove that the section of this trihedral angle by any plane perpendicular to any of the edges, at a point different from the vertex $S$, is a right tr... | First, note that if the angles of the tetrahedron $SABC$ are acute, then a plane perpendicular to any edge of the tetrahedron at a point different from $S$ intersects the other edges at points different from $S$, so the section of the tetrahedron by this plane is a triangle. For example, a plane passing through point $... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXVI - II - Problem 6
Let $ f(x) $ and $ g(x) $ be polynomials with integer coefficients. Prove that if for every integer value $ n $ the number $ g(n) $ is divisible by the number $ f(n) $, then $ g(x) = f(x) \cdot h(x) $, where $ h(x) $ is a polynomial. Show by example that the coefficients of the polynomial $ h(x) ... | In the solution to the preparatory task D of Series III, we proved that if polynomials $f$ and $g$ have integer coefficients, $\mathrm{st} f \leq \mathrm{st} g$, and for every natural number $n$, the number $f(n)$ is a divisor of the number $g(n)$, then there exists an integer $c$ such that $g(x) = c \cdot f(x)$. There... | proof | Algebra | proof | Yes | Yes | olympiads | false |
XIV OM - III - Task 6
Through the vertex of a trihedral angle, in which no edge is perpendicular to the opposite face, a line perpendicular to the opposite edge is drawn in the plane of each face. Prove that the three obtained lines lie in one plane. | Let $SM$, $SN$, and $SP$ denote the lines perpendicular to the edges $SA$, $SB$, and $SC$ of the tetrahedron $SABC$ and lying in the planes of the faces $BSC$, $CSA$, and $ASB$, respectively. Given that no edge of the tetrahedron is perpendicular to the opposite face, the lines $SM$, $SN$, and $SP$ are uniquely determi... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XLIII OM - I - Problem 2
In square $ABCD$ with side length $1$, point $E$ lies on side $BC$, point $F$ lies on side $CD$, the measures of angles $EAB$ and $EAF$ are $20^{\circ}$ and $45^{\circ}$, respectively. Calculate the height of triangle $AEF$ drawn from vertex $A$. | The measure of angle $ FAD $ is $ 90^\circ - (20^\circ + 45^\circ) = 25^\circ $. From point $ A $, we draw a ray forming angles of $ 20^\circ $ and $ 25^\circ $ with rays $ AE $ and $ AF $, respectively, and we place a segment $ AG $ of length $ 1 $ on it (figure 2).
From the equality $ |AG| =|AB| = 1 $, $ | \measureda... | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
LVIII OM - I - Problem 5
Given is an acute triangle $ ABC $, where $ \angle ACB=45^\circ $. Point $ O $ is the center of the circumcircle of triangle $ ABC $, and point $ H $ is the orthocenter of triangle $ ABC $. The line passing through point $ O $ and perpendicular to line $ CO $ intersects lines $ AC $ and $ BC $... | Let $D$ be the foot of the altitude from vertex $B$, and let $G$ be the point symmetric to point $H$ with respect to the line $AC$ (Fig. 2); segments $BG$ and $AC$ intersect at point $D$.
Notice that point $G$ lies on the circumcircle of triangle $ABC$. Indeed, due to the perpendicularities $BH \perp AC$ and $CH \perp ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXXI - I - Problem 7
On a sphere of radius 1, 13 points are chosen. Prove that among these points, there are two whose distance is less than $ \frac{2\sqrt{3}}{3} $. | Consider a regular dodecahedron inscribed in a sphere bounded by a given sphere. Among the radii of the sphere connecting certain points with the center, two of them intersect one face of the dodecahedron. The angle formed by these radii is no greater than the angle formed by the radii passing through two non-adjacent ... | \frac{2\sqrt{3}}{3} | Geometry | proof | Yes | Yes | olympiads | false |
I OM - B - Task 3
Someone has an unbalanced two-pan scale (i.e., a scale
whose arms are unequal). Wanting to weigh 2 kg of sugar, they proceed
as follows: they place a 1 kg weight on the left pan of the scale,
and pour sugar onto the right pan until it balances; then:
after emptying both pans, they place the 1 kg wei... | Let's denote the amount of sugar weighed for the first time by $ x $, and for the second time by $ y $. The lengths of the arms of the balance are $ p $ and $ q $, where $ p \neq q $.
According to the lever law (Fig. l) we have:
Therefore,
since
Answer: More than 2 kg of sugar was weighed in total. | More\than\2\ | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
IV OM - I - Problem 8
Through the point $ M $ given on the extension of side $ AB $ of triangle $ ABC $, draw a line intersecting sides $ AC $ and $ BC $ at points $ N $ and $ P $ in such a way that segments $ AN $ and $ BP $ are equal. | Analyze. Let's choose a point $M$ - for example - on the extension of a side of a given triangle beyond point $B$ (Fig. 28). Suppose that the line $m$ is the solution to the problem, i.e., that $AN = BP$.
Draw a line through point $B$ parallel to line $m$; it will intersect the side $AC$ of the triangle at some point $... | proof | Geometry | math-word-problem | Yes | Incomplete | olympiads | false |
III OM - II - Task 5
A vertical mast located on a tower is seen at the largest angle from such a point on the ground, the distance of which from the axis of the mast is $ a $; this angle equals the given angle $ \alpha $. Calculate the height of the tower and the height of the mast. | Let segment $AB$ (Fig. 31) represent a tower, and segment $BC$ - a mast. From point $P$ on the ground, the mast is seen at an angle $BPC$, which is an inscribed angle in a circle passing through points $B$, $C$, and $P$. This angle is larger the smaller the radius of that circle. Among all circles passing through point... | Geometry | math-word-problem | Yes | Yes | olympiads | false | |
XV OM - I - Problem 2
Prove the following two divisibility rules for 7:
a) A natural number $ N $ is divisible by 7 if and only if the number $ P $ formed in the following way is divisible by 7: multiply the first digit from the left of the number $ N $ by 3 and add the second digit; multiply the resulting number by 3... | Let $ a_0, a_1, \ldots, a_n $ denote the consecutive digits of the number $ N $ (from left to right). Then
a) According to the problem, the number $ P $ is the $(n+1)$-th term of the sequence
therefore
Since for a natural number $ k $
then each term on the right side of equation (1) is divisible by $ 7 $. The numbe... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XXIII OM - II - Zadanie 1
Udowodnić, że nie istnieją takie liczby rzeczywiste $ a, b, c $, $ x_1, x_2, x_3 $, że dla każdej liczby rzeczywistej $ x $
oraz $ x_1 \neq x_2 $, $ x_2 \neq x_3 $, $ x_3 \neq x_1 $, $ abc \neq 0 $.
|
Przypuśćmy, że istnieją liczby rzeczywiste $ a, b, c, x_1, x_2, x_3 $ spełniające warunki zadania. Mnożąc pierwszą równość (1) przez $ x $ i odejmując od niej równość drugą otrzymujemy
Podstawiając tu $ x = x_3 $ otrzymujemy, że $ a(x_3^3 - 1) = 0 $. Ponieważ $ a \ne 0 $, więc wynika stąd, że $ x_3^3 - 1 =0 $, c... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
XXIV OM - I - Problem 5
Real numbers $ x_1, x_2, \ldots, x_n $ satisfy the equation
let $ m $ be the smallest, and $ M $ - the largest of these numbers. Prove that $ x_1^2 + x_2^2 + \ldots + x_n^2 \leq -nmM $ and determine when equality holds. | Since $ m \leq x_1 \leq M $ for $ i = 1, 2, \ldots, n $, therefore
it follows that $ - x_i^2 + (m + M) x_i - mM \geq 0 $, where $ i = 1, 2, \ldots, n $.
Adding these inequalities side by side given $ x_1 + x_2 + \ldots + x_n = 0 $, we obtain
Equality will hold here if and only if equality holds in each of t... | proof | Inequalities | proof | Incomplete | Yes | olympiads | false |
XXXVII OM - I - Problem 11
We roll a die once, and then as many times as the number of dots that came up on the first roll. Calculate the expected value of the sum of the dots rolled (including the first roll). | Let $ X_0 $ denote the number of eyes rolled in the first throw. If $ X_0 = j $, then $ j $ additional throws are performed. Denote their results sequentially by $ X_1, \ldots, X_j $ and assume $ X_k = 0 $ for $ k > j $. The random variable considered in the task is then the sum $ X = \sum_{k=0}^6 X_k $.
Let us fix $ k... | notfound | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
XX OM - III - Problem 5
For what values of $ n $ does there exist a polyhedron with $ n $ edges? | A polyhedron with $ n $ edges exists if and only if $ n \geq 6 $ and $ n \ne 7 $.
Proof. a) An example of a polyhedron with $ n = 2m $ ($ m \geq 3 $) edges is a pyramid whose base is a polygon with $ m $ sides.
b) An example of a polyhedron with $ n = 2m+1 $ ($ m \geq 4 $) edges can be obtained using a pyramid with $ 2... | proof | Geometry | math-word-problem | Incomplete | Yes | olympiads | false |
XXI OM - II - Problem 4
Prove that if triangle $ T_1 $ contains triangle $ T_2 $, then the perimeter of triangle $ T_1 $ is not less than the perimeter of triangle $ T_2 $. | At the vertices of triangle $ T_2 $, we erect lines perpendicular to its sides (Fig. 11). These lines determine points $ A_1 $, $ B_1 $, $ B_2 $, $ C_2 $, $ C_3 $, $ A_3 $ on the boundary of triangle $ T_1 $. Since $ AB \bot AA_1 $ and $ AB \bot BB_1 $, the length of segment $ \overline{AB} $ is not greater than the le... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XLIX OM - III - Problem 4
Prove that in the sequence $ (a_n) $ defined by the formulas
there are infinitely many numbers divisible by $ 7 $.
Note: $ [n/2] $ is the greatest integer not exceeding $ n/2 $.
| In the sequence $(a_n)$, there are terms divisible by $7$ (for example, $a_5 = 7$). Suppose there are finitely many such terms; let $a_k$ be the last of these terms. Denote: $a_k = x$, $a_{2k-1} = y$, $a_{4k-3} = z$; the number $x$ is divisible by $7$, while the numbers $y$ and $z$ are not.
The following equalities hol... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
IV OM - III - Task 6
What algebraic relationship holds between $ \alpha $, $ \beta $, and $ \gamma $ when the equality is satisfied
翻译完成，但根据您的要求，最后一行似乎是不完整的。如果您需要进一步的信息或有其他任务，请告诉我。 | We can write equality (1) in the form
From this, it follows that
and after appropriate grouping of terms, we have
that is,
From the last equality, we conclude that | notfound | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
II OM - II - Task 1
In a right triangle $ ABC $ with the right angle at vertex $ C $, a height $ CD $ is drawn from vertex $ C $, and a circle is inscribed in each of the triangles $ ABC $, $ ACD $, and $ BCD $. Prove that the sum of the radii of these circles is equal to the height $ CD $. | We adopt the notations indicated in Figure 52a; let the radii of the circles inscribed in triangles $ABC$, $BCD$, $CAD$ be denoted by $r$, $r_1$, $r_2$, respectively.
Since the right triangles $ABC$, $CBD$, $ACD$ are similar, the radii of the circles inscribed in these right triangles are proportional to their hypotenu... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXXII - II - Problem 2
Two circles are internally tangent at point $ P $. A line tangent to one of the circles at point $ A $ intersects the other circle at points $ B $ and $ C $. Prove that line $ PA $ is the angle bisector of $ \angle BPC $. | om32_2r_img_11.jpg
Consider a homothety with center $ P $ and scale equal to the ratio of the radius of the larger circle to the radius of the smaller one (Fig. 11). This homothety transforms the smaller circle into the larger one, and the tangent $ BC $ into the line $ B_1C_1 $ tangent to the larger circle at point $ ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXX OM - I - Task 3
Numbers $ a, b, c, d, e $ are positive rational numbers. Prove that there exist rational numbers $ u_1, u_2, \ldots, u_n $ and non-negative rational numbers $ w_1, w_2, \ldots, w_n $, such that | Let $ A $ be the set of all non-zero polynomials with rational coefficients. A number $ \alpha $ is called algebraic if it is a root of some polynomial belonging to $ A $.
First, we will prove a few simple properties of algebraic numbers.
If $ \alpha $ is an algebraic number, then $ \alpha + 1 $ is also an algebraic nu... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
LX OM - I - Task 7
A sequence of integers $ f_0, f_1, f_2, \cdots $ is defined
by the conditions: $ f_0 = 0, f_1 = 1 $,
Find all polynomials $ W $ with integer coefficients,
having the following property: For every $ n =0, 1, 2, \cdots $
there exists an integer $ k $ such that $ W(k)= f_n $. | Answer: $ W(x)= \varepsilon x+c $, where $ \varepsilon \in \{-1,1\} $
and $ c $ is an integer.
We will repeatedly use the following fact: if $ F $
is a polynomial with integer coefficients, then for any
different integers $ u, v $, the divisibility
$ u-v|F(u)-F (v) $ holds.
To prove this fact, it suffices to note that ... | W(x)=\varepsilonx+ | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
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